* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

In mathematics, more specifically in functional analysis, a Banach space ( pronounced ) is a complete normed vector space.

With respect to this norm B ( X, Y ) is a Banach space.

Is X a Banach space, the space B ( X ) = B ( X, X ) forms a unital Banach algebra ; the multiplication operation is given by the composition of linear maps.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

* Theorem If X is a normed space, then X ′ is a Banach space.

If F is also surjective, then the Banach space X is called reflexive.

* Theorem Every reflexive normed space is a Banach space.

* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

* Corollary Let X be a reflexive normed space and Y a Banach space.

For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

The prototypical example of a Banach algebra is, the space of ( complex-valued ) continuous functions on a locally compact ( Hausdorff ) space that vanish at infinity.

* The algebra of all bounded real-or complex-valued functions defined on some set ( with pointwise multiplication and the supremum norm ) is a unital Banach algebra.

* The algebra of all bounded continuous real-or complex-valued functions on some locally compact space ( again with pointwise operations and supremum norm ) is a Banach algebra.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

Several elementary functions which are defined via power series may be defined in any unital Banach algebra ; examples include the exponential function and the trigonometric functions, and more generally any entire function.

( Gelfand – Naimark theorem ) Properties of the Banach space of continuous functions on a compact Hausdorff space are central to abstract analysis.

The space C of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm.

* Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.

The set C of continuous real-valued functions on, together with the supremum norm, is a Banach algebra, ( i. e. an associative algebra and a Banach space such that for all f, g ).

Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives.

* Functional analysis studies spaces of functions and introduces concepts such as Banach spaces and Hilbert spaces.

* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

In fact, the space of all Lipschitz functions on a compact metric space is dense in the Banach space of continuous functions, an elementary consequence of the Stone – Weierstrass theorem.

In particular the set of all real-valued Lipschitz functions on a compact metric space X having Lipschitz constant ≤ K is a locally compact convex subset of the Banach space C ( X ).

Topological rings occur in mathematical analysis, for examples as rings of continuous real-valued functions on some topological space ( where the topology is given by pointwise convergence ), or as rings of continuous linear operators on some normed vector space ; all Banach algebras are topological rings.

The set of all bounded analytic functions with the supremum norm is a Banach space.

These include theorems about compactness of certain spaces such as the Banach – Alaoglu theorem on the compactness of the unit ball of the dual space of a normed vector space, and the Arzelà – Ascoli theorem characterizing the sequences of functions in which every subsequence has a uniformly convergent subsequence.

Likewise, the Banach space C () of continuous functions on is not reflexive.

* C *- algebra: A Banach algebra that is a closed *- subalgebra of the algebra of bounded operators on some Hilbert space.

When the Banach algebra A is the algebra L ( X ) of bounded linear operators on a complex Banach space X ( e. g., the algebra of square matrices ), the notion of the spectrum in A coincides with the usual one in the operator theory.

* The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers C. Conversely, any compact subset of C arises in this manner, as the spectrum of some bounded linear operator.

* If is the norm ( usually noted as ) defined in the sequence space ℓ < sup >∞</ sup > of all bounded sequences ( which also matches the non-linear distance measured as the maximum of distances measured on projections into the base subspaces, without requiring the space to be isotropic or even just linear, but only continuous, such norm being definable on all Banach spaces ), and is lower triangular non-singular ( i. e., ) then

* The Banach space l < sup >∞</ sup > of all bounded real sequences, with the supremum norm, is not separable.

* Every separable metric space is isometric to a subset of the ( non-separable ) Banach space l < sup >∞</ sup > of all bounded real sequences with the supremum norm ; this is known as the Fréchet embedding.

The Stone – Čech compactification can be used to characterize ( the Banach space of all bounded sequences in the scalar field R or C, with supremum norm ) and its dual space.

In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum.

The space of bounded linear operators B ( X ) on a Banach space X is an example of a unital Banach algebra.

The spectrum of a bounded linear operator T acting on a Banach space X is the set of complex numbers λ such that λI − T does not have an inverse that is a bounded linear operator.

A bounded operator T on a Banach space is invertible, i. e. has a bounded inverse, if and only if T is bounded below and has dense range.

This extends the definition for bounded linear operators B ( X ) on a Banach space X, since B ( X ) is a Banach algebra.